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Analytic and differential geometry

Part of the Science Networks. Historical Studies book series (SNHS, volume 35)

Abstract

The two final chapters in volume I comprise a “complete theory of curves and curved surfaces”; that is, not only the “application of the differential calculus to the theory of curves” (and of curved surfaces) — what we now call differential geometry — but also the “purely algebraic part of that theory” — analytic geometry. Lacroix explained the inclusion of analytic geometry by his desire to offer a full set (“ensemble complet”) and to relate notions that were usually presented from very different points of view [Traité, I, xxv, 327].

Keywords

Tangent Plane Descriptive Geometry Algebraic Curf Plane Curf Conic Section 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 2.
    Belhoste [1992, 568] reads here “avec vous”; but given the teacher-pupil tone of the rest of the letter, this does not sound very convincing (unless of course Lacroix was writing that separate work as lectures for this student). Belhoste also interprets this whole passage as meaning that Lacroix intended to interpose his “descriptive geometry” [Lacroix 1795] in the Traité. I disagree: Lacroix certainly made many references to [Lacroix 1795] in chapter 5, but what he says here is that a work he had been writing on the application of analysis (to geometry, presumably) was going to be interposed in the Traité — that separate work must correspond to chapters 4 and 5.Google Scholar
  2. 4.
    In the 16th century the possibility of access to ancient Greek mathematical works had increased considerably because of the printing of both original versions and (usually Latin) translations. This (particularly the publication in 1588 of Commandino’s Latin translation of Pappos’ Mathematical Collection) had given origin to what Bos calls “the early modern tradition of geometrical problem solving” [Bos 2001, ch. 4].Google Scholar
  3. 5.
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  8. 12.
    Lacroix had published a textbook Traité élémentaire de trigonométie et d’application de l’algèbre à la géométrie [1798b], combining in one volume these two subjects; [Lacroix & Bézout 1826] was a combined translation of Lacroix’s trigonometry and Bézout’s application of algebra to geometry.Google Scholar
  9. 14.
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    Both Taton [1951, 135] and Boyer [1956, 220] wrongly ascribe this little priority to Jean-Baptiste Biot. Biot published in 1802 a Traité analytique des courbes et des surfaces du second degré; he changed the title of this work in the second edition (1805) to Essai de géométrie analytique, appliqué aux courbes et aux surfaces du second degré. Boyer had the excuse that he apparently did not see the first edition and assumed it had the same title as the second [Boyer 1956, 273]; but Taton [1951, 132] gave all these (and more) bibliographic details.Google Scholar
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  15. 32.
    To be more precise, it will be divisible by tn+1, where n is the largest integer by which it would be divisible in general (that is, the multiplicity of that point). This procedure can be found in [Cramer 1750, 460–464] and [Goudin & du Séjour 1756, 77–78]. Transformation of coordinates are fundamental tools in these books.Google Scholar
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    An interesting remark is that between the curve and any of these circles it is impossible to pass another circle [l’Hôpital 1696, 73]. It is interesting because Lagrange will use this property as a definition of contact.Google Scholar
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    Lagrange had also given these simplest osculating curves, but only as a comment, after having dealt with the general theory [Fonctions, 129–130].Google Scholar
  19. 51.
    It was common belief in the 18th century that all functions were piecewise monotonic. [Lagrange Fonctions, 155–156] for instance, has a similar assumption (also in a proof that the ordinate is the derivative of the area).Google Scholar
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    This characterization of envelopes can be seen for instance in [l’Hôpital 1696, ch. 8].Google Scholar
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  22. 60.
    For each point in a curve, the radius of curvature is the radius of an evolute, but for two consecutive points in a space curve, the radii of curvature are radii of different evolutes. In fact, there is an important exception to this rule: when the curve is a line of least or greatest curvature of a surface, its centres of curvature do form an evolute. Monge implicitly reported this in [1781, 690], stating that the normals are tangent to that curve, but apparently he never recognized explicitly that it is an evolute. Lagrange [Fonctions, 183], on the other hand, was quite explicit, and Hachette cited him in a footnote in [Monge & Hachette 1799, 357].Google Scholar
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    Although Monge [Feuilles, n° 7-i] had already used it in this sense, applied to surfaces. Lagrange [Fonctions] spoke of “courbes enveloppantes” and “surfaces enveloppantes”.Google Scholar
  27. 80.
    Which is not correct in general. [Coolidge 1940, 136] gives the example of any non-planar curve with prime order.Google Scholar

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